Before a change in career I used to work in the private client area of the law. In my experience from dealing with estates where the deceased was a BAEC member, BAEC always (and very kindly given the rules) transferred the balance of avios to the beneficiary under the Will (or under the intestacy rules if no will) provided the said beneficiary had their own BAEC account and of course once BA had seen the appropriate documentation.

A well crafted polite letter should do the trick.